158

ATTRACTIONS.

[ART. 309

where E is the mass of the earth; P2, P4, &c. are Legendre's functions and /3, 7 are two constants. We shall also suppose that the surface of the earth has the

form r=a(l-ecos'20-;p2sin20cos20)...........................(2),

where a is the semiaxis major and if the form be a spheroid, ^ = |e2.

If we substitute from (2) and (3) in (1) as in Art. 304 the result should be an identity. This will be found to be true if j8 and 7 are small quantities respectively of the first and second orders, and the expression for F in (3) is restricted to the first three terms. Equating to zero the coefficients of cos2 0 and cos40, (all the higher powers having coefficients of at least the third order), we thus obtain two equations to determine /3 and 7.

Let m be the ratio of the centrifugal force at the equator to equatorial gravity,

then w2a=wi ( - _-co2aj ,

where a is to be written for r after the differentiation has been performed, and cos 6 put equal to zero.

In this way we obtain the three results

2£"" .-" '--.^-fj^.....................(i),

..................................(5),

ay 35 a2

j?

ura = ~?tt{l + e-fw}

After substituting these values of /3 and 7 in (3) we have an expression for the potential of the earth at all external points. To find gravity g at the surface, wo have

J?

where V — V+ J w-r-sin-^. On substituting this valuo of F' we soon see that the expression for// contains terms which are constant multiples of cos20 and cos4^. Wo may therefore write

f/ = G'{l4-X<!os-tf H-A*sinu^coHa^} ........................... (8).

To find the three constants (/', X, /u. \ve notice that y~(f when ^ = ^TT. Hence G'is the value of equatorial gravity, and may be found from (7) by putting r = a and /? = ^7r after the differentiations have been performed. We observe next that \G' is the difference between the values of gravity at the pole and the equator and that loth these may be deduced from (7). Lastly we notice that - pG' is the coefficient of ens4/? in the value of tj ; and thin may be very shortly deduced from (7). In this

way we iind 6" .-_- ', . 1 - ;j m -f r - 'i I mt ~l~ -t m" + \ e" + if 2}"[ >

X ~ :: //< — e - 1 1 nit. - 2 e" •{- SJ j/~, //— '/'-7«c 4- 6" - !>/>'".

The angle 0 is the angle the radius vector r makes with the axis of rotation. If 6' be the angle the direction of gravity makes with the axis of rotation we have

0 = 0'+ lie sin tf'cos tf'. We then find by an easy substitution

<7 = G" { 1 + \ cos- #' -i- (M ..... 4\e) sin" 0' cos2 ^'}. We may extend Clairaut's theorem to a third approximation by proceeding in